Preface ........................................................ ix
1 The Basic Modular Forms of the Nineteenth Century ............ 1
1.1 The Modular Group ....................................... 1
1.2 Modular Forms ........................................... 5
1.3 Exercises .............................................. 11
2 Gauss's Contributions to Modular Forms ...................... 13
2.1 Early Work on Elliptic Integrals ....................... 13
2.2 Landen and Legendre's Quadratic Transformation ......... 17
2.3 Lagrange's Arithmetic-Geometric Mean ................... 18
2.4 Gauss on the Arithmetic-Geometric Mean ................. 20
2.5 Gauss on Elliptic Functions ............................ 27
2.6 Gauss: Theta Functions and Modular Forms ............... 32
2.7 Exercises .............................................. 36
3 Abel and Jacobi on Elliptic Functions ....................... 42
3.1 Preliminary Remarks .................................... 42
3.2 Jacobi on Transformations of Orders 3 and 5 ............ 54
3.3 The Jacobi Elliptic Functions .......................... 60
3.4 Transformations of Order n and Infinite Products ....... 63
3.5 Jacobi's Transformation Formulas ....................... 66
3.6 Equivalent Forms of the Transformation Formulas ........ 70
3.7 The First and Second Transformations ................... 71
3.8 Complementary Transformations .......................... 72
3.9 Jacobi's First Supplementary Transformation ............ 74
3.10 Jacobi's Infinite Products for Elliptic Functions ...... 75
3.11 Jacobi's Theory of Theta Functions ..................... 80
3.12 Jacobi's Triple Product Identity ....................... 86
3.13 Modular Equations and Transformation Theory ............ 89
3.14 Exercises .............................................. 90
4 Eisenstein and Hurwitz ...................................... 94
4.1 Preliminary Remarks .................................... 94
4.2 Eisenstein's Theory of Trigonometric Functions ........ 101
4.3 Eisenstein's Derivation of the Addition Formula ....... 105
4.4 Eisenstein's Theory of Elliptic Functions ............. 106
4.5 Differential Equations for Elliptic Functions ......... 109
4.6 The Addition Theorem for the Elliptic Function ........ 113
4.7 Eisenstein's Double Product ........................... 115
4.8 Elliptic Functions in Terms of the ф Function ......... 116
4.9 Connection of Ø with Theta Functions .................. 117
4.10 Hurwitz's Fourier Series for Modular Forms ............ 123
4.11 Hurwitz's Proof That Δ(ω) Is a Modular Form ........... 126
4.12 Hurwitz's Proof of Eisenstein's Result ................ 128
4.13 Kronecker's Proof of Eisenstein's Result .............. 129
4.14 Exercises ............................................. 130
5 Hermite's Transformation of Theta Functions ................ 132
5.1 Preliminary Remarks ................................... 132
5.2 Hermite's Proof of the Transformation Formula ......... 138
5.3 Smith on Jacobi's Formula for the Product of Four
Theta Functions ....................................... 141
5.4 Exercises ............................................. 147
6 Complex Variables and Elliptic Functions ................... 149
6.1 Historical Remarks on the Roots of Unity .............. 149
6.2 Simpson and the Ladies Diary .......................... 161
6.3 Development of Complex Variables Theory ............... 164
6.4 Hermite: Complex Analysis in Elliptic Functions ....... 172
6.5 Riemann: Meaning of the Elliptic Integral ............. 176
6.6 Weierstrass's Rigorization ............................ 182
6.7 The Phragmén-Lindelöf Theorem ......................... 184
7 Hypergeometric Functions ................................... 188
7.1 Preliminary Remarks ................................... 188
7.2 Stirling .............................................. 189
7.3 Euler and the Hypergeometric Equation ................. 191
7.4 Pfaff's Transformation ................................ 192
7.5 Gauss and Quadratic Transformations ................... 193
7.6 Kummer on the Hypergeometric Equation ................. 196
7.7 Riemann and the Schwarzian Derivative ................. 198
7.8 Riemann and the Triangle Functions .................... 201
7.9 The Ratio of the Periods K'/K as a Conformal Map ...... 202
7.10 Schwarz: Hypergeometric Equation η with Algebraic
Solutions ............................................. 207
7.11 Exercises ............................................. 210
8 Dedekind's Paper on Modular Functions ...................... 212
8.1 Preliminary Remarks ................................... 212
8.2 Dedekind's Approach ................................... 216
8.3 The Fundamental Domain for SL2 ( ) .................... 219
8.4 Tesselation of the Upper Half-plane ................... 222
8.5 Dedekind's Valency Function ........................... 222
8.6 Branch Points ......................................... 223
8.7 Differential Equations ................................ 225
8.8 Dedekind's η Function ................................. 228
8.9 The Uniqueness of k2 .................................. 234
8.10 The Connection of η with Theta Functions .............. 234
8.11 Hurwitz's Infinite Product for η(ω) ................... 235
8.12 Algebraic Relations among Modular Forms ............... 236
8.13 The Modular Equation .................................. 238
8.14 Singular Moduli and Quadratic Forms ................... 243
8.15 Exercises ............................................. 249
9 The η Function and Dedekind Sums ........................... 251
9.1 Preliminary Remarks ................................... 251
9.2 Riemann's Notes ....................................... 258
9.3 Dedekind Sums in Terms of a Periodic Function ......... 264
9.4 Rademacher ............................................ 269
9.5 Exercises ............................................. 274
10 Modular Forms and Invariant Theory ......................... 276
10.1 Preliminary Remarks ................................... 276
10.2 The Early Theory of Invariants ........................ 279
10.3 Cayley's Proof of a Result of Abel .................... 285
10.4 Reduction of an Elliptic Integral to Riemann's
Normal Form ........................................... 287
10.5 The Weierstrass Normal Form ........................... 289
10.6 Proof of the Infinite Product for Δ ................... 291
10.7 The Multiplier in Terms of 12√Δ ....................... 293
11 The Modular and Multiplier Equations ....................... 295
11.1 Preliminary Remarks ................................... 295
11.2 Jacobi's Multipfier Equation .......................... 303
11.3 Sohnke's Paper on Modular Equations ................... 304
11.4 Brioschi on Jacobi's Multiplier Equation .............. 314
11.5 Joubert on the Multiplier Equation .................... 317
11.6 Kiepert and Klein on the Multiplier Equation .......... 320
11.7 Hurwitz: Roots of the Multiplier Equation ............. 326
11.8 Exercises ............................................. 332
12 The Theory of Modular Forms as Reworked by Hurwitz ......... 334
12.1 Prefiminary Remarks ................................... 334
12.2 The Fundamental Domain ................................ 335
12.3 An Infinite Product as a Modular Form ................. 336
12.4 The J-Function ........................................ 339
12.5 An Application to the Theory of Elliptic Functions .... 342
13 Ramanujan's Euler Products and Modular Forms ............... 344
13.1 Preliminary Remarks ................................... 344
13.2 Ramanujan's τ Function ................................ 348
13.3 Ramanujan: Product Formula for Δ ...................... 350
13.4 Proof of Identity (13.2) .............................. 353
13.5 The Arithmetic Function τ(n) .......................... 356
13.6 Mordell on Euler Products ............................. 362
13.7 Exercises ............................................. 367
14 Dirichlet Series and Modular Forms ......................... 371
14.1 Preliminary Remarks ................................... 371
14.2 Functional Equations for Dirichlet Series ............. 373
14.3 Theta Series in Two Variables ......................... 380
14.4 Exercises ............................................. 382
15 Sums of Squares ............................................ 384
15.1 Preliminary Remarks ................................... 384
15.2 Jacobi's Elliptic Functions Approach .................. 393
15.3 Glaisher .............................................. 394
15.4 Ramanjuan's Arithmetical Functions .................... 397
15.5 Mordell: Spaces of Modular Forms ...................... 400
15.6 Hardy's Singular Series ............................... 405
15.7 Hecke's Solution to the Sums of Squares Problem ....... 410
15.8 Exercises ............................................. 424
16 The Hecke Operators ........................................ 426
16.1 Preliminary Remarks ................................... 426
16.2 The Hecke Operators T(n) .............................. 428
16.3 The Operators T(n) in Terms of Matrices λ(n) .......... 434
16.4 Euler Products ........................................ 438
16.5 Eigenfunctions of the Hecke Operators ................. 439
16.6 The Petersson Inner Product ........................... 442
16.7 Exercises ............................................. 444
Appendix: Translation of Hurwitz's Paper of 1904 .............. 445
§1. Equivalent Quantities .................................. 445
§2. The Modular Forms Gn(ω1, ω2) ........................... 448
§3. The Representation of the Function Gn by Power Series .. 452
§4. The Modular Form Δ(ω1, ω2) ............................. 454
§5. The Modular Function J(ω) .............................. 455
§6. Applications to the Theory of Elliptic Functions ....... 460
Bibliography .................................................. 463
Index ......................................................... 471
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